As a more specific instance of this question, we can consider for \(V,\varepsilon >0\)  the subgroup \(\Theta _{V,\varepsilon }\subset \Theta _3\)  generated by the integral homology spheres with volume \(\le V\)  and injectivity radius \(\ge \varepsilon \) . As the latter is a finite set [1]}, \(\Theta _{V,\varepsilon }\)  is a finitely generated hence proper subgroup of \(\Theta _3\) . Furthermore, by Myers' theorem we have that
\(\bigcup _{V,\varepsilon >0}\Theta _{V,\varepsilon }=\Theta _3,\) 
